RE: question on finiteness in toposes

[categories <cat-dist@mta.ca>]

Date: Wed, 15 Jan 97 10:19 GMT
From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk>

Not an answer to Bill's question (which I agree is an important one),
but a minor correction. Bill wrote:

While the K/S definition is right for the construction of
the object classifier over an arbitrary base topos (as Gavin
showed) and hence for classifiers for various kinds of
finitary algebras over an arbitrary base topos,

It isn't, and he didn't. Gavin used finite cardinals to construct
the object classifier over an arbitrary base topos with NNO (and I
subsequently extended the construction to finitary algebraic
theories), but it doesn't work over a topos without NNO (and in
particular it can't be made to work using K-finiteness). Andreas
Blass showed that the existence of an object classifier for toposes
over E implies that E has a NNO.

Incidentally, I think it is correct to give credit to Kuratowski for
the notion of K-finiteness. It's true that Sierpinski's paper was
earlier, but his definition was a "global" one (i.e. he defined the
class of all finite sets as the sub-semilattice of the universe
generated by he singletons), whereas Kuratowski made the crucial
observation that the finiteness of a particular set X can be determined
locally (i.e. within the power-set of X), without which the notion
could never have been imported into topos theory.

Peter Johnstone